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Properties

Label 2-672-7.5-c2-0-11
Degree $2$
Conductor $672$
Sign $0.0608 - 0.998i$
Analytic cond. $18.3106$
Root an. cond. $4.27909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (3.60 + 2.08i)5-s + (3.89 + 5.81i)7-s + (1.5 − 2.59i)9-s + (3.55 + 6.16i)11-s − 2.34i·13-s − 7.21·15-s + (9.39 − 5.42i)17-s + (20.0 + 11.5i)19-s + (−10.8 − 5.35i)21-s + (−6.35 + 11.0i)23-s + (−3.82 − 6.61i)25-s + 5.19i·27-s − 2.42·29-s + (−16.7 + 9.69i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (0.721 + 0.416i)5-s + (0.555 + 0.831i)7-s + (0.166 − 0.288i)9-s + (0.323 + 0.560i)11-s − 0.180i·13-s − 0.481·15-s + (0.552 − 0.319i)17-s + (1.05 + 0.609i)19-s + (−0.517 − 0.255i)21-s + (−0.276 + 0.478i)23-s + (−0.152 − 0.264i)25-s + 0.192i·27-s − 0.0836·29-s + (−0.541 + 0.312i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0608 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0608 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.0608 - 0.998i$
Analytic conductor: \(18.3106\)
Root analytic conductor: \(4.27909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1),\ 0.0608 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.851625272\)
\(L(\frac12)\) \(\approx\) \(1.851625272\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (-3.89 - 5.81i)T \)
good5 \( 1 + (-3.60 - 2.08i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3.55 - 6.16i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 2.34iT - 169T^{2} \)
17 \( 1 + (-9.39 + 5.42i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-20.0 - 11.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (6.35 - 11.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 2.42T + 841T^{2} \)
31 \( 1 + (16.7 - 9.69i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-2.81 + 4.87i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 32.1iT - 1.68e3T^{2} \)
43 \( 1 - 0.536T + 1.84e3T^{2} \)
47 \( 1 + (-2.50 - 1.44i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-26.9 - 46.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (72.8 - 42.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-81.5 - 47.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-44.0 - 76.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 62.2T + 5.04e3T^{2} \)
73 \( 1 + (45.3 - 26.2i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (42.1 - 73.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 77.3iT - 6.88e3T^{2} \)
89 \( 1 + (123. + 71.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 38.0iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.37381520044288079799042597912, −9.763335341764668908548216475384, −8.995717974301700019265794937465, −7.83203104266614466984704632024, −6.91021608997303562772192964540, −5.70918117226822605620010878477, −5.39997121798235298868709729099, −4.05291672545301325572034424065, −2.69088378616173031840532096120, −1.46405069529457281808674053370, 0.77748717710232078811951857595, 1.80921141574753231153097730052, 3.50331404473511937759229465482, 4.74627069266497908189464458408, 5.54570460252551063784835927288, 6.47293381197749016686480137267, 7.43252750546382765338579892953, 8.275102650987832293441254364490, 9.366528643559142719378168116333, 10.08225486213424567807449970382

Graph of the $Z$-function along the critical line